Mathematical logic seminar - Apr 14 2020

Time: 3:30pm - 4:30 pm

Room: https://cmu.zoom.us/j/621951121

Speaker:     Maxwell Levine    
University of Vienna

Title: Patterns of stationary reflection

Abstract:

Singular cardinals yield surprising results in set theory. After Cohen proved that CH is independent of ZFC Easton proved that on regular cardinals, the continuum function κ ↦ 2κ is constrained only by the facts that λ ≤ κ implies 2λ ≤ 2κ and that cf(2κ) > κ. In other words, the ZFC constraints on κ ↦ 2κ are fully characterized relative to the class of regular cardinals. In an unexpected turn, Silver proved that GCH cannot fail for the first time at a singular cardinal of uncountable cofinality. In other words, the failure of CHκ is ``compact'' for such cardinals. More constraints on the arithmetic of singular cardinals were later discovered by Shelah using his PCF theory.

We are not limited to studying cardinal arithmetic when considering these compactness phenomena. We can also investigate the compactness of □κ, a canonical property of Gödel's Constructible Universe L. Cummings, Foreman, and Magidor showed that it is consistent for □n, to hold for all n < ω while □ω fails. However, the question for singulars of uncountable cofinality is still open.

We will present an Easton-style result for stationary reflection, which is relevant because failure of stationary reflection at κ+ is an important consequence of □κ. If S is a stationary subset of a cardinal κ, the reflection principle SR(S) asserts that every stationary subset of S reflects. Assuming the consistency of a supercompact cardinal, we prove that given a fixed n<ω, there are only a few trivial ZFC constraints on SR(κ ∩ cof(ℵn) (current work in inner model theory suggests that the large cardinal assumption is close to optimal). The successors of singular cardinals present the greatest hurdle for this result, and require a nonstandard usage of PCF theory.